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The traditional model of mathematics teaching for science students goes a bit like this: first we teach them a set of mathematical concepts and skills, and then the students "apply" those concepts and skills to their subjects. This works fine some of the time and for some students, but it has a well-known drawback, namely, that for many students the mathematics stays forever separate from the science and never gets "applied" at all. Moreover, the way science uses mathematics is changing. At one time the mathematics that scientists used was generally explicit but often, outside certain disciplines, routine. Scientists in many fields now use techniques of analysis that reflect increasingly sophisticated mathematical models, but these models are often implicit in the technology of the lab. The scientist of the 21st century needs to have access to, and understanding of, the mathematical models that underlie her work: she needs that just as much as, if not more than, mastery of traditional pen-and-paper algorithms.
Here at Imperial, the departments of mathematics and chemistry are working on a joint teaching project based on Mathematica. The governing philosophy of the project is that students should, from early on in their first year, encounter math as an integral part of chemistry and that they should be encouraged to examine, amend, devise, and reflect upon mathematical and statistical models of chemical systems. Without computer power, such a project would have foundered because without computer power the students would have access only to trivial, tractable models that don't adequately describe the science. With the wrong kind of computer power, it would have foundered for another reason: if, for example, we had merely written a collection of "simulations," then the mathematics would have remained hidden behind an opaque user interface.
What Mathematica gives us is power plus explicitness: the ability to set up relatively sophisticated models in a way that allows students to ungroup and amend them and then to set up their own. The typesetting features of the new front end have helped us to get students into the mathematics quickly and smoothly without needing first to master an arcane syntax. I will demonstrate some examples of the materials we are devising at Imperial, together with some preliminary findings from a systematic evaluation of them in use that is being carried out by the Institute of Education in London.
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